Crossword Genius

In a Big Apple chain coffee bar, leave tip no matter how much (2,3,4)

Ross

I believe the answer is:

at any cost

I'm a little stuck... Click here to teach me more about this clue!

'much' is the definition.
I know nothing about this answer so I can't tell whether it can be defined by this definition.

'in a big apple chain coffee bar leave tip no matter' is the wordplay.
I cannot quite see how this works, but
'a' could be 'an' and 'an' is found within the answer.
'big' could be 'os' and 'os' is found in the answer.
'matter' could be 'at' (At is an example) and 'at' is found in the answer.
This explanation may well be incorrect...

'how' is the link.

Can you help me to learn more?

(Other definitions for at any cost that I've seen before include "Regardless of the price to pay" , "whatever the damage" .)

I've seen this clue in The Independent.
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